Scientific visualization involves displaying large sets of data generated in the process of investigating a scientific or engineering problem. In many engineering applications, large linear systems of equations arise. Such applications include, for example, simulation of nonlinear microwave circuits, robotic control, or power systems design. Linear systems of equations are typically expressed in the formAx=b,  (1)where AεRn×n or AεCn×n, i.e., A is a square n-dimensional matrix with real or complex elements, xεRn×1 and bεRn×1, or xεCn×1 and bεCn×1, i.e., x and b are either real or complex n-dimensional vectors. The entries of the matrix A are values that arise from the physical system involved. For example, the matrix A is often a Jacobian matrix related to a nonlinear algebraic equation associated with a physical system, e.g., a conductance matrix of an electronic circuit, where x is a voltage vector and b is a current vector.
A matrices characterize both numerical and structural properties of the physical system. Numerical or quantitative properties are indicated by the elements of A being, for example, “real,” “complex,” “positive,” “negative,” “integer,” or “floating point” to name just a few. Structural or qualitative properties are related to the position or indexing of the elements of the matrix. The position of an element in a matrix is indexed by integers indicating the row and column position of the element. Note that matrices are 2-dimensional arrays and some physical systems may involve higher dimensional arrays such as 3-dimensional arrays. In the case of a 3-dimensional array, for example, the position of elements in the array are indexed by a row, a column, and a plane. Characterization of structural properties of a matrix include, for example: dense matrices, in which many of the elements of the matrix are non-zero; sparse matrices, in which many of the elements of the matrix are zero; Hermitian or symmetric matrices, in which a matrix equals its adjoint; and skew-symmetric matrices, in which the elements of the transpose of the matrix have the opposite sign from the elements of the matrix.
Many numerical methods exist for the solution of systems of equations expressed in the form of equation (1); see, for example, Kendall E. Atkinson, “An Introduction to Numerical Analysis,” (Wiley, 1989), and Yousef Saad, “Iterative Methods for Sparse Linear Systems,” (PWS Publishing Company, 1996). Numerical methods for the solution of these systems of equations include reordering, preconditioning, factoring, and substitution. The choice of numerical method for the solution of equation (1) is guided by the numerical and structural characteristics of the physical systems modeled.
There are conventional software packages that allow a user to store and display data arrays, such as matrices. For example, commercial software packages such as MATLAB® by the Math Works Inc. or MATHEMATICA® by Wolfram Research. These software packages, however, may not be able to display a conductance matrix associated with the operation of an electrical circuit with millions of components. The main difficulty with conventional packages is the limitation placed on allocation of memory provided for the storage of data arrays, thus limiting the ability of conventional packages to display large arrays. The deficiencies of the conventional software packages show that a need still exists for a method and system which permit visualization of large sets of data.